Lyapunov exponents for surface groups representations

TL;DR

This paper constructs a bifurcation current for holomorphic families of surface group representations into PSL(2,C), linking Lyapunov exponents, Brownian motion, and the distribution of bifurcation phenomena on the parameter space.

Contribution

It introduces a new bifurcation current defined via Lyapunov exponents and Brownian motion, revealing the asymptotic distribution of bifurcations in the parameter space.

Findings

Bifurcation current describes asymptotic distribution of bifurcations.

Random hypersurfaces are asymptotically equidistributed with respect to the bifurcation current.

The approach combines Lyapunov exponents, Brownian motion, and discretization techniques.

Abstract

Let (\rho_\la)_{\la\in \La} be a holomorphic family of representations of a surface group \pi_1(S) into PSL(2,C), where S is a topological (possibly punctured) surface with negative Euler characteristic. Given a structure of Riemann surface of finite type on S we construct a bifurcation current on the parameter space \La, that is a (1,1) positive closed current attached to the bifurcations of the family. It is defined as the $dd^c$ of the Lyapunov exponent of the representation with respect to the Brownian motion on the Riemann surface S, endowed with its Poincare metric. We show that this bifurcation current describes the asymptotic distribution of various codimension 1 phenomena in \La. For instance, the random hypersurfaces of \La defined by the condition that a random closed geodesic on S is mapped under \rho_\la to a parabolic element or the identity are asymptotically equidistributed with respect to the bifurcation current. The proofs are based on our previous work "Random walks, Kleinian groups, and bifurcation currents", and on a careful control of a discretization procedure of the Brownian motion.